讨论具有不同自变量的变系数模型的函数系数的估计及其大样本性质使用局部线性方法和积分方法，得到函数系数的积分估计由于该估计有较大的方差，进一步使用回切法改进这一估计，获得了函数系数的改进估计同时，研究了改进估计的渐近正态性最后，用模拟例子说明提出的估计方法是有效的。

本文探讨具有不同光滑变量的变系数模型的建模、估计和估计的渐近性。首先，从实际出发建立模型；然后，使用局部线性方法给出模型中未知函数的初始估计，再使用平均方法，给出它们的平均估计；进一步，给出这些平均估计的渐近正态性。两个模拟例子说明这一估计方法是有效的。

It is well known that the exposure to ambient air pollution might cause serious respiratory illnesses and that the weather conditions may also contribute to the seriousness. However, quantifying the effects of pollution and the weather condition is a difficult task due to the nonlinear nature of these impacts. The problem is further complicated by the possibly cumulative effects of these impacts. In this paper, the nonparametric additive (NPA) models, which have the advantage of ease in interpretation and forecasting, are employed for modeling the effects of pollution and weather. All models are derived by the local linear method. The variables in the final selected NPA model are chosen by cross-validation method together with bootstrap test for the data of Hong Kong. For comparison the final selected linear regression (LR) model by the backward elimination method is also considered. It is found, interestingly, that the variables selected by nonparametric method and the usual backward elimination method for linear models are different. Furthermore, by comparing forecasted values obtained from the NPA and LR models and true values the final selected NPA model is shown to outperform the LR model.

The functional-coefficient regression models with different smoothing variables in different coefficient functions are discussed in this paper. The averaged estimates of coefficient functions are defined by averaging the sample on the initial value obtained by a local linear technique. Their asymptotic normality is studied. The efficiency of the proposed method is shown by a simulated example.

In this paper, we are concerned with statistical inference for the index parameter α0 in the single-index model Y=g (αT0X) + ε. Based on the estimates obtained by the local linear method, we extend the generalized likelihood ratio test to the single-index model. We investigate the asymptotic behaviour of the proposed test and demonstrate that its limiting null distribution follows a 2-distribution, with the scale constant and the number of degrees of freedom being independent of nuisance parameters or functions, which is called the Wilks phenomenon. A simulated example is used to illustrate the performance of the testing approach.